In a bit of political gamesmanship, Senate Majority Leader Harry Reid floated the idea of using the House GOP's border supplemental bill as a vehicle to enact comprehensive immigration reform.

"Well, if they pass that, maybe it's an opening for us to have a conference on our comprehensive immigration reform bill. They're finally sending us something on immigration; maybe we could do that," the Nevada Democrat told reporters on Tuesday.

Asked if that prospect would make the House legislation likely to fail, given that the House GOP adamantly opposes the Senate-passed immigration bill, Reid said: "I'm not threatening anything. We've been looking for something to do conference on. Maybe we could do it with that."

Reid's remarks may complicate House Republican leaders' task of winning support for their border legislation, which they want to pass before leaving town at the end of the week for summer recess. House Speaker John Boehner's (R-OH) communications director, Kevin Smith, hastened to point out that GOP leaders have ruled out that prospect.

The House bill to address the influx of Central American children on the southern border, unveiled Tuesday, contains $659 million for border security and to speed up processing of the 57,000 estimated children apprehended within the last year. It also amends a 2008 anti-trafficking law to clarify that minors coming from all countries can be quickly sent back home.

Senate Minority Whip John Cornyn (R-TX) on Tuesday said Republicans won't allow Democrats to use humanitarian crisis legislation as a vehicle for comprehensive immigration reform. He reiterated his opposition to simply "throw[ing] money at the problem" without amending border laws. He expressed frustration with Reid's remarks, suggesting the Democratic leader was trying to poison the House GOP's border bill.

"It's a totally phony threat. And I think people need to understand that," Cornyn told reporters. "I think [Reid is] trying to get the House to back down."

Congress is highly unlikely to agree to a supplemental in time for August recess, so the lingering question is whether one party can pass its own bill for political cover with constituents. Democrats would rather not go home and explain why the Republican-led House passed a bill to address the border crisis but the Democratic-led Senate failed to.

In a statement shortly after Reid's remarks, Boehner accused the Democratic leader of launching a "deceitful and cynical attempt to derail" the House GOP's bill.

"So let me be as clear as I can be with Senator Reid: the House of Representatives will not take up the Senate immigration reform bill or accept it back from the Senate in any fashion," Boehner said. "Nor will we accept any attempt to add any other comprehensive immigration reform bill or anything like it, including the DREAM Act, to the House’s targeted legislation, which is meant to fix the actual problems causing the border crisis."